Download Slide 1

RijksUniversiteit Groningen
Nanoscience TopMaster 2006 Symposium
Negative Refraction
Asem Ampoumogli
1582542
Groningen June 2006
1
Refraction
• Refraction is one of the fundamental
phenomena in the interaction of matter and
radiation along with reflection, absorption
and scattering.
• When a ray of light traveling through
medium A (e.g. air) of refractive index n1
arrives at a medium B (e.g. glass) of
refractive index n2, it is refracted at the
interface.
• All kinds of waves refract i.e. acoustic
waves, seismic waves, even sea waves!
2
Index of refraction
• The refractive index n is defined by the equation:
n   r r
• Where:
•εr is the relative electrical permittivity, defined as εr=ε/ε0
•μr is the relative magnetic permeability, defined as μr =μ/μ0
Both these constants are generally positive numbers.
3
Index of refraction
• In 1621 the Dutch mathematician Willebrord Snell arrived at the sinθ form of
the law of refraction.
n1 sin 1  n2 sin 2
• Upon entering medium B, the radiation’s
phase velocity will change to:
c
up 
n
The refractive index is always positive.
4
Veselago’s classic paper
In 1968 the Russian physicist Victor Veselago published a paper in which he
studied the interaction of electromagnetic radiation with a (hypothetical)
material for which both ε and μ are simultaneously negative.
V.G. Veselago, Soviet Physics USPEKHI, 10, 509 (1968).
He arrived at some interesting conclusions:
• A planar slab of this material will focus
light (not parallel rays though).
• The vector set k, E, H will be left handed,
which means the group and the phase
velocities in this material will be in opposite
directions.
•The energy flow E×H of an
electromagnetic wave will be in the opposite
direction to the wave vector because the
permeability is negative.
http://sagar.physics.neu.
.edu/lhm-intro-1.html
5
Veselago’s classic paper
Most importantly, Veselago was able to show that this material would have a
negative index of refraction.
i
i
n    exp( )  exp( )  exp(i )  1
2
2
This has some more interesting
consequences:
• The Doppler shift will be reversed.
• The Cerenkov radiation will be reversed.
• Snell’s law will be reversed.
How can this happen? What does it mean?
Why are there no materials readily found that
exhibit a negative refractive index?
6
The Drude –Lorentz model and
negative response
In the Drude-Lorentz atomic model, the atoms are modeled as oscillators (the electrons
are bound to the nucleus with a spring) with a resonant frequency ω0.
An electric field of frequency ω will drive the oscillations of the system at ω.
If the frequency of the radiation is near the resonant frequency the induced polarization
becomes very large.
In the classical analogue (mass on an ideal spring), below the resonant frequency, the
mass is displaced in the same direction as the driving force (analogy: mass=dipole
moment).
However, above the resonant frequency, the mass is displaced in a direction opposite to
the driving force.
In the case of resonance with the electric field, the material will exhibit a negative
permittivity ε (as is the case with metals at optical frequencies).
At resonance with a driving magnetic field, the material will exhibit a negative
permeability .
So, the negative index of refraction will be observed when the material is in
resonance with both the electric field and the magnetic field.
7
Negative Index materials
Can we create materials that will have a negative index even for a narrow
frequency window?
In 1999 Professor J. Pendry of the London Imperial college published a paper
proposing structures that, was predicted, would feature the needed
electromagnetic properties to have a measurable negative refractive index.
J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, IEEE transactions on microwave theory
and techniques, 47, 2075 (1999).
The proposed structures are to be created with two elements:
Cut wires (designed to have a specific electrical resonance frequency) and
Split ring resonators (SRR’s) (designed to have a specific magnetic resonance
frequency).
8
Negative Index materials
Because the radiation that will be used has a wavelength hundreds of times
larger than the atomic units in our material, the atomic details lose importance
in describing how the material will interact with it.
It was further argued that we can define an effective permittivity and an
effective permeability to macroscopically describe our materials.
From this point of view we will have created a meta-material whose unique
properties are not determined by the fundamental physical properties of its
constituents but by the shape and the distribution of the specific patterns
included in them.
What would something like that look like?
9
Permittivity
 p2  02
 eff ( )  1  2
  02  i
The resonant frequency can be set to virtually
any value in this kind of materials, so the
negative ε can be reproduced at low
frequencies rather than just the optical region.
Above ω0 and below ωp the effective
permittivity is negative.
10
Permeability
The split ring resonator (SRR) will feature a
magnetic response without being inherently
magnetic.
We will have the meta-material equivalent of a
magnetic atom.
F 2
eff ( )  1  2
  02  i
0  LC  1/ LC
11
Other designs
12
Experimental verification
R. A. Shelby, D. R. Smith, & S. Schultz, Science 292, 77-79 (2001).
The experiment was carried out at a
frequency of 10.5 GHZ.
The control sample was made of
Teflon, cut with a step pattern
identical to that of the LHM sample
Results:
NIM: n= - 2.7
Teflon: n= 1.4
13
Applications
The most prominent application of refractive materials is in the manufacturing
of lenses.
Veselago (as well as Pendry) have shown that NIM lenses would not require
curved surfaces to focus radiation (but would not focus parallel rays and will
have a magnification of unity).
14
Applications
One of the most dramatic – and controversial- prediction for the NIMs was that
by Pendry in 1999 which stated that a thin negative-index film should behave as
a “superlens”, providing image detail with a resolution beyond the diffraction
limit, to which all positive-index lenses are subject.
 k z  k0 cos  

k x  k0 sin  

k0
This restriction is a significant problem.


2
15
Applications
Magnetic Resonance Imaging (MRI):
Large quasi-static fields cause the nuclear spins in the patient’s body to align. The spins are resonant at
the local Larmor frequency, so a second magnetic field (an RF field) will excite them causing them to
precess around the magnetic field. Images are constructed by observing the time dependent signal
resulting from the precession of the spins.
16